Feature learning via mean-field Langevin dynamics: classifying sparse parities and beyond.

Neural network in the mean-field regime is known to be capable of \textit{feature learning}, unlike the kernel (NTK) counterpart. Recent works have shown that mean-field neural networks can be globally optimized by a noisy gradient descent update termed the \textit{mean-field Langevin dynamics} (MFLD). However, all existing guarantees for MFLD only considered the \textit{optimization} efficiency, and it is unclear if this algorithm leads to improved \textit{generalization} performance and sample complexity due to the presence of feature learning. To fill this gap, in this work we study the statistical and computational complexity of MFLD in learning a class of binary classification problems. Unlike existing margin bounds for neural networks, we avoid the typical norm control by utilizing the perspective that MFLD optimizes the \textit{distribution} of parameters rather than the parameter itself; this leads to an improved analysis of the sample complexity and convergence rate. We apply our general framework to the learning of $k$-sparse parity functions, where we prove that unlike kernel methods, two-layer neural networks optimized by MFLD achieves a sample complexity where the degree $k$ is ``decoupled'' from the exponent in the dimension dependence.